\documentstyle[11pt,psfig]{article}
\newcommand{\bvec}[1]{\mathbf{#1}}
\begin{document}
\title{\Large\bf Random Collage Model with Scale Invariance}
\author{Ann B. Lee and David Mumford}
\date{}
\maketitle
\subsection*{\centering Abstract}
{\em We present a model for scale invariance of natural images
based on the ideas of images as collages of statistically independent
objects. The model takes occlusions into account, and produces images that
show translational invariance, and approximate scale invariance under
block averaging and median filtering. We compare the
statistics of the simulated images with data from natural scenes, and
find good agreement for short-range and middle-range statistics.
Furthermore, we discuss the implications of the model on a 3D
description of the world.}
\begin{figure}[htb]
\begin{tabular}{cc}
\mbox{\psfig{figure={sample6.eps},width=2.5in}} &
\mbox{\psfig{figure={scaling.eps},width=2.5in}}
\end{tabular}
\caption{\footnotesize
{\bf Left:} Sample from the random collage model.
The image has $256 \times 256$ pixels, and is constructed from
opaque discs with size distribution $f(r) \sim 1/r^{3}$, where
the radius $1/2 \leq r \leq 8 \cdot 256 = 2048$ pixels.
{\bf Right:} Log of marginal distributions
of $\nabla_{x} I^{\sigma}$ for scales $\sigma=2,4,8,16$.
Scaling is done by median filtering $2 \times 2$ blocks recursively,
and the ensemble average is taken over 1000 images.}
\end{figure}
\end{document}